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May 5, 2012

Twisted Higher Bundles in Münster

Posted by Urs Schreiber

Right now I am at the 17th NRW Topology Meeting. In a few minutes I will talk about Principal ∞-Bundles – Theory and Applications.

By coincidence it turns out that the previous speaker, Ulrich Pennig discussed, in a nice talk, such an application: twisted 2-vector bundles.

This is joint work of him and Brano Jurčo. They consider BDR 2-vector bundles which, by definition, are the objects classified by, roughly, the monoidal delooping of the monoidal category GL (Vect)GL_\bullet(Vect). Their starting point to consider twists of these structures is the discussion in Thomas Kragh’s Orientations and Connective Structures on 2-vector Bundles, who constructs, for each nn, a fiber sequence

BOGl n(Vect)BGl n(Vect)cK( 2,3), B OGl_n(Vect) \to B Gl_n(Vect) \stackrel{c}{\to} K(\mathbb{Z}_2, 3) \,,

and interprets the space on the left as that classifying “oriented BDR 2-vector bundles”, in higher analogy of orientation of vector bundle. Accordingly, the map cc induces a notion of twisted (twisted oriented) 2-vector bundles, with twist a class in H 3(, 2)H 4(,)H^3(-,\mathbb{Z}_2) \hookrightarrow H^4(-, \mathbb{Z}), hence with twisting \infty-bundles specific B 2U(1)B^2 U(1)-principal 3-bundles (aka bundle 2-gerbes).

This is similar to the twisted String 2-bundles which are twisted by the fractional Pontryagin class in H 4(,)H^4(-, \mathbb{Z}).

Have to run now. More details later.

Posted at May 5, 2012 10:34 AM UTC

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